bounded sequence as convergent

electronic engineer

Some rule says that not all bounded sequence must be convergent sequence , one example is the sequence with general bound:
Xn=(-1)^n

could anyone help?!
thanks in advance!

Muzza

What is it that you want help with? To show that x_n is divergent, it suffices to find two subsequences of x_n which converge to different numbers.

matt grime

Ooh, erm, let me think... how about looking at the bloody definitions? Yes, I am tired, so delete as applicable, but quite frankly, I've seen enough of this for one day.

fourier jr

if a sequence converges then it's bounded, but if it's bounded it might not converge as that example shows.

electronic engineer

you're right, the question is how to prove that,could you help?!

matt grime

What is the negation of: x_n converges to x? Show that this is satisfied. Of course if x_n converges to x then all subsequences of x_n converge to x as well, maknig for an easy proof that it doesn't converge. If you're not used to working out the negations of propositions then say so.

fourier jr

suppose that the sequence {a_n} converges. then a_n --> L for some finite L. now look at the definition of convergence & see that for ANY epsilon (yadda yadda)... for n>N but to make things easier just choose epsilon=1 & if n>N get 2 inequalities |a_n| < |a_n - L| + |L| < 1 + |L| (i think that's right, you check it). if n<N then |a_n| < something else.

so for any n>0, |a_n| < M where M=max{a certain set of numbers}. end of proof

HallsofIvy

Prove what? That there exist bounded sequences that are not convergent? You do that by exhibiting one, just as you did.

Prove that that particular sequence is not convergent? Just as Muzza said in the very first response to your post: show that there exist two different subsequences that converge to two different limits- in this case the subsequence with n even: 1, 1, 1, ... converges to 1, the subsequence with n odd: -1, -1, -1, ... converges to -1. Since arbitrarily far into the sequnce there exist numbers arbitrarily close to 1 and numbers arbitrarily close to -1, taking [itex]\epsilon= 1/3[/itex] will show that no number can be the limit of the entire sequence.